`PQ` is an infinite current carrying conductor. `AB` and `CD` are smooth conducting rods on which a conductor `EF` moves with constant velocity `v` as shown. The force needed to maintain constant speed of `EF` is

Options

A. `1/(VR)[(mu_0IV)/(2pi) ln (b)/(a)]^2`
B. `v/R[(mu_0IV)/(2pi)ln(a)/(b)]^2`
C. `v/R[(mu_0IV)/(2pi)ln(b)/(a)]^2`
D. None of these

Answer

A

Explanation

`B_x=mu_0/(2pi)I/x`
`de=B_x vdx=mu_0/(2pi) I/x v dx`
`e=int_a^bde(mu_0Iv)/(2pi) ln (b/a)`
`i=e/R=(mu_0Iv)/(2pir)ln(b/a)=`induced current
`dF=(i)(dx)B_x`
`[(mu_0Iv)/(2piR)ln(b/a)][mu_0/(2pi)I/x]dx`
`F=int_a^bdF`